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A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

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We are concerned with the Umemura polynomials associated with rational solutions of the third Painlevé equation. We extend Taneda's method, which was developed for the Yablonskii–Vorob'ev polynomials associated with the second Painlevé equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation which determines the Umemura polynomials are indeed polynomials. Our proof is constructive and gives information about the roots of the Umemura polynomials.
Keywords: Umemura polynomials; third Painlevé equation; recurrence relation. 
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     author = {Peter A. Clarkson and Chun-Kong Law and Chia-Hua Lin},
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Peter A. Clarkson; Chun-Kong Law; Chia-Hua Lin. A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a79/

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